MULTISCALE ANALYSIS OF 1-RECTIFIABLE MEASURES: NECESSARY CONDITIONS

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1 MULTISCAL ANALYSIS OF 1-RCTIFIABL MASURS: NCSSARY CONDITIONS MATTHW BADGR AND RAANAN SCHUL Abstract. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R n, n 2. To each locally finite Borel measure µ, we associate a function J 2 (µ, x) which uses a weighted sum to record how closely the mass of µ is concentrated near a line in the triples of dyadic cubes containing x. We show that J 2 (µ, ) < µ-a.e. is a necessary condition for µ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to 1-dimensional Hausdorff measure. 1. Introduction The aim of this article is to develop a multiscale analysis of 1-rectifiable measures. Because there exist competing conventions for the terminology rectifiable measure (cf. [Fed69, pp ] and [Mat95, p. 228]), we start by specifying its meaning in the present paper. See Table 1.1 for a guide between the different conventions. Definition 1.1 (Rectifiable measure). Let µ be a Borel measure on R n and let m 1 be a positive integer. We say that µ is m-rectifiable if there exist countably many bounded Borel sets i R m and Lipschitz maps f i : i R n such that the union of the images f i ( i ) have full measure, i.e. µ (R n \ i=1 f i( i )) = 0. Remark 1.2. We do not require an m-rectifiable measure µ to be absolutely continuous with respect to the m-dimensional Hausdorff measure H m (see 2). If we wish to declare this as an additional property of the measure, then we shall explicitly write µ H m. Remark 1.3. An equivalent condition for a Borel measure µ on R n to be 1-rectifiable is that there exist countably many rectifiable curves Γ i R n such that µ(r n \ i Γ i) = 0. (Indeed, any Lipschitz map f : R n, R m extends to a global Lipschitz map F : R m R n. Thus, when m = 1, one may assume without loss of generality that the sets i in Definition 1.1 are compact intervals [a i, b i ] and take Γ i = f i ([a i, b i ]).) The qualitative theory of rectifiable sets and absolutely continuous rectifiable measures in uclidean spaces developed across the last century, beginning with the seminal work of Date: September 15, Mathematics Subject Classification. Primary 28A75. Key words and phrases. rectifiable measure, singular measure, Jones beta number, Jones function, Analyst s Traveling Salesman Theorem, Hausdorff density, Hausdorff measure and packing measure. M. Badger was partially supported by an NSF postdoctoral fellowship DMS R. Schul was partially supported by a fellowship from the Alfred P. Sloan Foundation as well as by NSF DMS

2 2 MATTHW BADGR AND RAANAN SCHUL Table 1.1. Cross-reference: rectifiable measures this paper [Fed69] [Mat95] µ is m-rectifiable R n is countably (µ, m) rectifiable µ is m-rectifiable and µ(r n ) < R n is (µ, m) rectifiable µ is m-rectifiable and µ H m µ is m-rectifiable Besicovitch [Bes28, Bes38] and later generalized and improved upon in a series of papers by Morse and Randolph [MR44], Moore [Moo50], Marstrand [Mar64], Mattila [Mat75] and Preiss [Pre87]. In particular, in the presence of absolute continuity, these investigations revealed a deep connection between the rectifiability of a measure and the asymptotic behavior of the measure on small balls. Definition 1.4 (Hausdorff densities). Let B(x, r) denote the closed ball in R n with center x R n and radius r > 0. For each positive integer m 1, let ω m = H m (B m (0, 1)) denote the volume of the unit ball in R m. For all locally finite Borel measures µ on R n, we define the lower Hausdorff m-density D m (µ, ) and upper Hausdorff m-density D m (µ, ) by and D m (µ, x) := lim inf r 0 D m (µ, x) := lim sup r 0 µ(b(x, r)) ω m r m [0, ] µ(b(x, r)) ω m r m [0, ] for all x R n. If D m (µ, x) = D m (µ, x) for some x R n, then we write D m (µ, x) for the common value and call D m (µ, x) the Hausdorff m-density of µ at x. For µ a Borel measure and R n a Borel set, let µ denote the restriction of µ to, i.e. the measure defined by the rule (µ )(F ) = µ( F ) for all Borel F R n. Theorem 1.5 ([Mat75]). Let 1 m n 1. Suppose R n is Borel and µ = H m is locally finite. Then µ is m-rectifiable if and only if the Hausdorff m-density of µ exists and D m (µ, x) = 1 at µ-a.e. x R n. Theorem 1.6 ([Pre87]). Let 1 m n 1. If µ is a locally finite Borel measure on R n, then µ is m-rectifiable and µ H m if and only if the Hausdorff m-density of µ exists and 0 < D m (µ, x) < at µ-a.e. x R n. There exist additional characterizations of absolutely continuous rectifiable measures (e.g. in terms of the tangent measures of µ). For a full survey, we refer the reader to the book [Mat95] by Mattila. In general, m-rectifiable measures on R n are not necessarily absolutely continuous with respect to Hausdorff measure H m. In the case m = 1, an interesting family of singular 1-rectifiable measures was recently identified by Garnett, Killip and Schul [GKS10]. xample 1.7 ([GKS10]). Let h : R R be the 1-periodic function defined by { 2 if x [ 1 h(x) =, 2) + Z otherwise

3 MULTISCAL ANALYSIS OF 1-RCTIFIABL MASURS 3 Given 0 < δ 1/3, define the measure η on R to be the weak- limit of measures η k, k 1 dη k := [1 + (1 3δ)h(3 j x)] dx. j=0 Finally, assign µ = η η to be the n-fold product of η. When δ = 1/3, the measure µ is Lebesgue measure on R n. On the other hand, Garnett, Killip and Schul proved that if δ δ n < 1/3 is sufficiently small, then µ is 1-rectifiable in the sense of Definition 1.1 (see Theorem 1.1 and the paragraph In closing... on p of [GKS10]). Therefore, for all n 2, there exist locally finite Borel measures µ on R n with the following properties. The support of µ is R n : µ(b(x, r)) > 0 for all x R n and for all r > 0. The measure µ is doubling: there is C > 1 such that µ(b(x, 2r)) Cµ(B(x, r)) for all x in the support of µ and for all r > 0. very Lipschitz graph (i.e. a set which up to an isometry of R n has the form {(x, f(x)) : f : R m R n m is Lipschitz} for 1 m n 1) has µ measure zero. For every connected Borel set Γ R n, the set {x Γ : D 1 (H 1 Γ, x) < } has µ measure zero. The measure µ is 1-rectifiable. The measure µ is singular with respect to H 1. In our opinion, general rectifiable measures that are allowed to be singular with respect to Hausdorff measure are currently poorly understood. We believe that the following open problem represents a major challenge in geometric measure theory. Problem 1.8. For all 1 m n 1, find necessary and sufficient conditions in order for a locally finite Borel measure µ on R n to be m-rectifiable. (Do not assume that µ H m.) In this paper, we adapt tools from the theory of quantitative rectifiability to attack Problem 1.8 in the case m = 1. In particular, we establish new necessary conditions for a locally finite Borel measure µ on R n to be 1-rectifiable. To state these results, we need to introduce two concepts L 2 beta numbers and L 2 Jones functions which emanate from [Jon90, DS91, DS93, BJ94] (also see [Paj96, Paj97, Lég99, Ler03, DT12]). For all R n and x R n, let diam := sup y,z y z and dist(x, ) := inf z F x z denote respectively the diameter of and the distance of x to. Definition 1.9 (L 2 beta numbers). For every locally finite Borel measure µ on R n and every bounded Borel set Q R n (typically we take Q to be a cube), define β 2 (µ, Q) by ( ) 2 dist(x, l) (1.1) β2(µ, 2 dµ(x) Q) := inf [0, 1], l diam Q µ(q) Q where l in the infimum ranges over all lines in R n. If µ(q) = 0, then we interpret (1.1) as β 2 (µ, Q) = 0. The beta number β 2 (µ, Q) records how well µ Q is fit by a linear regression model, in an L 2 sense. At one extreme, β 2 (µ, Q) = 0 if and only if µ Q = µ (Q l 0 ) for some line l 0 in R n. At the other extreme, β 2 (µ, Q) 1 when the mass of µ Q is scattered in the sense that µ Q assigns non-negligible mass far away from every line passing through Q.

4 4 MATTHW BADGR AND RAANAN SCHUL One may think of several natural ways to add up the errors β 2 (µ, Q) at all scales. Below we focus on two variations. Let (R n ) denote the standard dyadic grid; that is, the collection of all (closed) dyadic cubes in R n. Also, for each cube Q and λ > 0, let λq denote the concentric cube about Q that is obtained by dilating Q by a factor of λ. Definition 1.10 (L 2 Jones functions). Let µ be a locally finite Borel measure on R n and let r > 0. The ordinary L 2 Jones function J 2 (µ, r, ) for µ is defined by J 2 (µ, r, x) := Q β 2 2(µ, 3Q)χ Q (x) [0, ] for all x R n, where Q ranges over all Q (R n ) with side length at most r. abbreviate the function J 2 (µ, 1, ) starting at scale 1 by J 2 (µ, ). The density-normalized L 2 Jones function J 2 (µ, r, ) for µ is defined by (1.2) J2 (µ, r, x) := Q β 2 2(µ, 3Q) diam Q µ(q) χ Q(x) [0, ] We for all x R n, where Q ranges over all Q (R n ) with side length at most r. (Here we take 0/0 = 0.) We abbreviate the function J 2 (µ, 1, ) starting at scale 1 by J 2 (µ, ). The ordinary L 2 Jones function J 2 (µ, ) has been used by several authors to study the rectifiability of absolutely continuous measures of the form µ = H 1, R n. Although we formulate xample 1.11, Theorem 1.12 and Theorem 1.13 for 1-rectifiable measures only, analogous statements for m-rectifiable measures exist for all m 2. xample 1.11 ([DS91]). Let R n be Borel and 1-Ahlfors regular, i.e. suppose that there exist constants c 1, c 2 > 0 such that c 1 r H 1 ( B(x, r)) c 2 r for all x and for all 0 < r < diam. If µ = H 1 is 1-uniformly rectifiable in the sense of David and Semmes [DS91, DS93], then there exists a constant C = C(µ) > 0 such that J Q 2(µ, diam Q, ) dµ C diam Q for every dyadic cube Q. In particular, J 2 (µ, ) < µ-a.e. Theorem 1.12 ([Paj97, Theorem 1.1]). Suppose K R n is compact and µ = H 1 K is finite. If both D 1 (µ, x) > 0 and J 2 (µ, x) < at µ-a.e. x R n, then µ is 1-rectifiable. Theorem 1.13 ([Paj97, Theorem 1.2]). If K R n is compact and 1-Ahlfors regular, then µ = H 1 K is 1-rectifiable if and only if J 2 (µ, ) < µ-a.e. By contrast, for singular rectifiable measures, J 2 (µ, ) can be badly behaved. xample Let µ be a measure from xample 1.7 with defining parameter δ δ n. Since β 2 (µ, 3Q) 1 for every dyadic cube Q, the ordinary Jones function J 2 (µ, ) = µ-a.e. Nevertheless, it follows from the estimates in [GKS10] or by Theorem A below that the density-normalized Jones function J 2 (µ, ) < µ-a.e. In 2000, Peter Jones conjectured that weighted L 2 Jones functions should lead to a solution of Problem 1.8 (private communication). Nam-Gyu Kang obtained unpublished results about this conjecture for measures supported on the four-corner Cantor set in R 2 (private communication). The general case is more complicated, as is evident by the measures in xample 1.7.

5 MULTISCAL ANALYSIS OF 1-RCTIFIABL MASURS 5 We are now ready to state our main results. Theorem A confirms Jones conjecture by connecting the rectifiability of a measure to the pointwise behavior of its densitynormalized L 2 Jones function. Corollary B shows that for absolutely continuous measures, rectifiability of a measure also controls its ordinary L 2 Jones function. Theorem A. Let µ be a locally finite Borel measure on R n. If µ is 1-rectifiable, then J 2 (µ, x) < at µ-a.e. x R n. Corollary B. Let µ be a locally finite Borel measure on R n. If µ is 1-rectifiable and µ H 1, then J 2 (µ, x) < at µ-a.e. x R n. In the special case of 1-dimensional Hausdorff measure restricted to a compact set, Corollary B (together with Lemma 2.7) immediately yields the converse to Theorem Corollary C. Suppose that K R n is compact and µ = H 1 K is finite. Then µ is 1-rectifiable if and only if both D 1 (µ, ) > 0 and J 2 (µ, ) < µ-a.e. x R n. A qualitative consequence of Theorem A is that a 1-rectifiable measure µ exhibits at least one of two extreme behaviors at µ-almost every x in R n : µ admits arbitrarily good 1- dimensional linear approximations near x or µ has arbitrarily large 1-dimensional density ratio near x. Thus, if one extreme fails, the other extreme must occur. xample Let µ be a measure from xample 1.7 with defining parameter δ δ n. Since J 2 (µ, x) < and β(µ, 3Q) 1 for every dyadic cube Q, the Hausdorff 1-density of µ exists and D 1 (µ, x) = at µ-a.e. x R n. The proofs of Theorem A and Corollary B will be given in 2 (prerequisites) and 3 (main arguments). We remark that because the proofs rely on the Traveling Salesman Theorem for rectifiable curves in R n, it is not immediately clear how to use our method to study m-rectifiable measures for m 2. Our work also leaves open the possibility of finding sufficient conditions for rectifiability. Nevertheless, we believe that the idea encoded in the definition of J 2 (µ, ) to normalize a multiscale quantity by the density of a measure scale-by-scale is a fruitful idea that should prove useful in additional situations. To end the paper, in 4, we discuss some connections between Theorem A and Corollary B, and prior work of Léger [Lég99] (Menger curvature), Lerman [Ler03] (curve learning) and Tolsa [Tol12] (mass transport). Acknowledgements. The authors would like to thank Marianna Csörnyei for insightful discussions about this project. The authors would also like to thank an anonymous referee for his or her careful reading of the paper. Part of this work was carried out while both authors visited the Institute for Pure and Applied Mathematics (IPAM) during the Spring 2013 long program on Interactions between Analysis and Geometry. 2. Traveling Salesman Theorem and Other Prerequisites In this section, we recall an essential tool from the theory of quantitative rectifiability: the Analyst s Traveling Salesman Theorem. We also collect miscellaneous lemmas which facilitate the proofs in section 3.

6 6 MATTHW BADGR AND RAANAN SCHUL Definition 2.1. Let R n be any set. Q, define the quantity β (Q) by β (Q) := inf l For every bounded set Q R n such that dist(x, l) sup x Q diam Q where l ranges over all lines in R n. By convention, we set β (Q) = 0 if Q =. Theorem 2.2 (Traveling Salesman Theorem, [Jon90, Oki92]). A bounded set R n is a subset of a rectifiable curve in R n if and only if β 2 () := β (3Q) 2 diam Q <. Moreover, there exists a constant C = C(n) (1, ) (independent of ) such that β 2 () CH 1 (Γ) for every connected set Γ containing, and there exists a connected set Γ such that H 1 (Γ) C(diam + β 2 ()). Corollary 2.3. For all n 2 and 3 < a <, there is a constant C = C (n, a) (1, ) such that if R n is bounded and Γ is a connected set containing, then β (aq) 2 diam Q C H 1 (Γ). Proof. Let n 2 and let 3 < a < be given. For any dyadic cube Q (R n ) and integer k 0, let Q k (R n ) denote the kth ancestor of Q, i.e. Q k is the unique dyadic cube containing Q with diam Q k = 2 k diam Q. Choose m 1 large enough depending only on a such that aq 3Q m for all Q (R n ) (e.g. m = log 2 a will suffice.) Then β (aq) (3 2 m /a)β (3Q m ) 2 m β (3Q m ) for all R n and Q (R n ). Hence β (aq) 2 diam Q 2 2m β (3Q m ) 2 diam Q = 2 m = 2 m(1+n) β (3Q m ) 2 diam Q m β (3Q) 2 diam Q 2 m(1+n) CH 1 (Γ) whenever R n is bounded and Γ is a connected set containing by Theorem 2.2. Remark 2.4. Inspecting the proof in [Oki92] shows that the constants C in Theorem 2.2 depends exponentially on the dimension n. Schul [Sch07b] formulated a version of the Analysts Traveling Salesman Theorem that is valid in infinite-dimensional Hilbert space. However, to obtain Theorem 2.2 with constants that are independent of the dimension n, one must replace the grid of dyadic cubes appearing in the definition of β 2 () with a multiresolution family of balls that are adapted to a sequence (X k ) k=1 of 2 k -nets for the set. We now enumerate several lemmas that will be used in section 3, starting with some facts from geometric measure theory. To fix conventions, we recall the definitions of Hausdorff and packing measures in R n. See [Mat95, Chapters 4 6] for general background.

7 MULTISCAL ANALYSIS OF 1-RCTIFIABL MASURS 7 Definition 2.5 (Hausdorff and packing measures in R n ). Let s 0 be a real number. Let, 1, 2,... denote Borel sets in R n. The s-dimensional Hausdorff measure H s is defined by H s () = lim δ 0 Hδ s () where { Hδ() s = inf (diam i ) s : } i, diam i δ. i i The s-dimensional packing premeasure P s is defined by P s () = lim δ 0 Pδ s () where { } Pδ s () = sup (2r i ) s : x i, 2r i δ, i j B(x i, r i ) B(x j, r j ) =. i The s-dimensional packing measure P s is defined by { P s () = inf P s ( i ) : = i i Lemma 2.6. Let µ be a locally finite Borel measure on R n. Then µ H s if and only if D s (µ, ) < µ-a.e. Lemma 2.6 is xercise 4 in [Mat95, Chapter 6]. Lemma 2.7. Let µ be a locally finite Borel measure on R n. If µ is m-rectifiable, then D m (µ, ) > 0 µ-a.e. To prove Lemma 2.7, we first prove an auxiliary lemma. Lemma 2.8. Let R m. If f : R n is Lipschitz, then P s (f()) (Lip f) s P s () and P s (f()) (Lip f) s P s (). Proof. Assume that P s () < and f : R n is L-Lipschitz. Given ε > 0, pick η > 0 such that P s η () P s () + ε. Fix 0 < δ Lη and let {B n (f(x i ), r i ) : i 1} be an arbitrary disjoint collection of balls in R n centered in f() such that 2r i δ for all i 1. Since f is L-Lipschitz, i } f(b m (x i, r i /L)) B n (f(x i ), r i ) for all i 1. Thus {B m (x i, r i /L) : i 1} is a disjoint collection of balls in R m centered in such that 2r i /L δ/l η. Hence (2r i ) s = L s (2r i /L) s L s Pη s () L s (P s () + ε). i=1 i=1 Taking the supremum over all δ-packings of f(), we obtain Pδ s(f()) Ls (P s () + ε). Therefore, letting δ 0 and ε 0, P s (f()) L s P s (). The corresponding inequality for the packing measure P s follows immediately from the inequality for P s. Proof of Lemma 2.7. Let µ be a locally finite m-rectifiable Borel measure on R n and let A = {x R n : D m (µ, x) = 0}. To show that µ(a) = 0 it suffices to prove that µ(a f()) = 0 for every bounded set R m and every Lipschitz map f : R n, since µ is m-rectifiable. To that end fix R m bounded and f : R n Lipschitz..

8 8 MATTHW BADGR AND RAANAN SCHUL Then P m (f()) (Lip f) m P m () < by Lemma 2.8 and the assumption that R m is bounded. Let λ > 0. Since D m (µ, x) = 0 λ for all x A f(), we have µ(a f()) λp m (A f()) λp m (f()) by [Mat95, Theorem 6.11]. Thus, letting λ 0, µ(a f()) = 0 for every bounded set R m and every Lipschitz map f : R n. Therefore, µ(a) = 0, or equivalently, D m (µ, x) > 0 for µ-a.e. x R n. Next we make some comparisons between the different L 2 Jones functions defined in the introduction. Lemma 2.9. For every locally finite Borel measure µ, the sets {x R n : J 2 (µ, r, x) < } and {x R n : J 2 (µ, r, x) = }, and {x R n : J 2 (µ, r, x) < } and {x R n : J 2 (µ, r, x) = } are independent of the parameter r > 0. Proof. With µ and x fixed, changing the value of r > 0 inserts or deletes at most a finite number of terms in the defining sums for J 2 (µ, r, x) and J 2 (µ, r, x). Below side Q := diam Q/ n denotes the side length of a cube in R n. Lemma Let µ be a locally finite Borel measure and let x R n. If D 1 (µ, x) < and J 2 (µ, x) <, then J 2 (µ, x) <. Proof. Let µ be a locally finite Borel measure on R n, and assume that at some x R n, both D 1 (µ, x) < and J 2 (µ, x) <. Since D 1 (µ, x) < there exist constants M < and r 0 > 0 such that µ(b(x, r)) Mr for all 0 < r r 0. In particular, µ(q) µ(b(x, diam Q)) M diam Q for every cube Q containing x with side Q r 0 / n (that is, diam Q r 0 ). Hence J 2 (µ, r 0 / n, x) = β2(µ, 2 3Q)χ Q (x) (2.1) side Q r 0 / n M side Q r 0 / n β 2 2(µ, 3Q) diam Q µ(q) χ Q(x) = M J 2 (µ, r 0 / n, x). But J 2 (µ, r 0 / n, x) < by Lemma 2.9, since J 2 (µ, x) <. Hence J 2 (µ, r 0 / n, x) < by (2.1). Therefore, since J 2 (µ, r 0 / n, x) <, we have J 2 (µ, x) < by Lemma Proofs of the Main Results The proof of Theorem A is based on Proposition 3.1. Roughly speaking, this proposition says that if the lower density of a finite measure ν is uniformly bounded away from 0 along a subset of a rectifiable curve Γ R n, then J 2 (ν, ) has finite norm in L 1 (ν). In particular, J 2 (ν, x) < at ν-a.e. x.

9 MULTISCAL ANALYSIS OF 1-RCTIFIABL MASURS 9 Proposition 3.1. Let ν be finite Borel measure on R n and let Γ be a rectifiable curve. If Γ is Borel and there exists a constant c > 0 such that ν(b(x, r)) c r for all x and for all 0 < r r 0, then J 2 (ν, r 0, x) dν(x) H 1 (Γ) + ν(r n \ Γ), where the implied constant depends only on n and c (see (3.10)). Proof of Proposition 3.1. Let ν be a finite Borel measure on R n and let Γ be a rectifiable curve in R n. Assume that Γ is Borel and there exists a constant c > 0 such that ν( B(x, r)) c r for all x and for all 0 < r r 0. Our objective is to find an upper bound for J 2 (ν, r 0, x) dν(x) in terms of the ambient dimension n, the lower Ahlfors regularity constant c, the length of Γ, and ν(r n \ Γ). For the duration of the proof, we let a > 3 and ε > 0 denote fixed constants, depending on at most n, which will be specified after (3.5). To proceed divide the dyadic cubes (R n ) into three subfamilies 0, Γ and 2, as follows: 0 = {Q (R n ) : side Q > r 0 or ν( Q) = 0}, Γ = {Q (R n ) : side Q r 0, ν( Q) > 0 and εβ 2 (ν, 3Q) β Γ (aq)}, and 2 = {Q (R n ) : side Q r 0, ν( Q) > 0 and β Γ (aq) < εβ 2 (ν, 3Q)}. Thus, the family 0 consists of all of the dyadic cubes in R n J that do not contribute to 2 (ν, r 0, x) dν(x). And, of the remainder, the families Γ and 2 consist of the cubes for which either β Γ (aq) or εβ 2 (ν, 3Q) is the dominant quantity, respectively. Reading off the definitions of J 2 (ν, r 0, ), 0, Γ and 2, it follows that (3.1) J 2 (ν, r 0, x) dν(x) = β2(ν, 2 3Q) diam Q χ Q (x) dν(x) ν(q) Q = β 2 ν( Q) 2(ν, 3Q) diam Q ν(q) Q Γ 2 ε 2 + β2(ν, 2 3Q) diam Q. Q 2 Q Γ β 2 Γ(aQ) diam Q }{{} I } {{ } II We shall estimate the terms I and II separately. The former will be controlled by H 1 (Γ) and the latter will be controlled by ν(r n \ Γ). To estimate I, we note that by Corollary 2.3, (3.2) I ε 2 βγ(aq) 2 diam Q C ε 2 H 1 (Γ), where C is a finite constant determined by n and a. In order to estimate II, decompose R n \ Γ into a family T of Whitney cubes with the following specifications. The union over all sets in T is R n \ Γ. ach set in T is a half-open cube in R n of the form (a 1, b 1 ] (a n, b n ]. If T 1, T 2 T, then either T 1 = T 2 or T 1 T 2 =.

10 10 MATTHW BADGR AND RAANAN SCHUL If T T, then dist(t, Γ) diam T 4 dist(t, Γ). (To obtain this decomposition, modify the standard Whitney decomposition in Stein [St] by replacing each closed cube with the corresponding half-open cube.) Here dist(t, Γ) = inf x T inf y Γ x y. For each k Z, we define T k = {T T : 2 k 1 < dist(t, Γ) 2 k }. Also for every cube Q, we set T (Q) = {T T : ν(q T ) > 0} and T k (Q) = T k T (Q). Our plan is to first estimate β 2 2(ν, 3Q) diam Q for each Q 2 and then estimate II. Fix Q 2, say with side Q = 2 k 0 r 0. We remark that if T T k (3Q), then k k 1 (k 0 ) := k 0 1 log 2 3 n (to derive this, bound the distance between a point in T 3Q and a point in Q by diam 3Q). Pick any line l in R n such that (3.3) sup dist(z, l) 2β Γ (aq) diam aq < 2εβ 2 (ν, 3Q) diam aq z Γ aq = (2/3)aεβ 2 (ν, 3Q) diam 3Q. To control β 2 2(ν, 3Q), we divide 3Q into two sets, N ( near ) and F ( far ), where and N = {x 3Q : dist(x, l) (2/3)aεβ 2 (ν, 3Q) diam 3Q} F = {x 3Q : dist(x, l) > (2/3)aεβ 2 (ν, 3Q) diam 3Q}. It follows that ( ) 2 ( ) 2 dist(x, l) β2(ν, 2 dν(x) dist(x, l) 3Q) N diam 3Q ν(3q) + dν(x) F diam 3Q ν(3q) ( ) 2 dist(x, l) (2/3) 2 a 2 ε 2 β2(ν, 2 dν(x) (3.4) 3Q) + F diam 3Q ν(3q) ( ) 2 dist(x, Γ aq) (3.5) 3(2/3) 2 a 2 ε 2 β2(ν, 2 dν(x) 3Q) + 2 diam 3Q ν(3q), where to pass between (3.4) and (3.5) we used the triangle inequality, (3.3), and the inequality (p + q) 2 2p 2 + 2q 2. Note that dist(x, Γ) diam 3Q = 3 n side Q for all x F, because F 3Q and Γ Q (since ν( Q) > 0 for all Q 2 ). Hence, specifying a = 3+6 n and 3(2/3) 2 a 2 ε 2 = 1/2 (or generally a 3+6 n and ε 3/2a 6) ensures that dist(x, Γ aq) = dist(x, Γ) for all x F and β2(ν, 2 3Q) 4 F We now employ the Whitney decomposition. F ( ) 2 dist(x, Γ) dν(x) diam 3Q ν(3q). Since Γ 3Q N (by (3.3)), we have F T T (3Q) T 3Q = k=k 1 (k 0 ) T T k (3Q) T 3Q. Note that if T T k(3q) and x T, then dist(x, Γ) dist(t, Γ) + diam T 5 dist(t, Γ) 5 2 k. Thus, ( ) 2 (3.6) β2(ν, 2 k 2 ν(t 3Q) 3Q) 100. diam 3Q ν(3q) k=k 1 (k 0 ) T T k (3Q)

11 MULTISCAL ANALYSIS OF 1-RCTIFIABL MASURS 11 To continue, recall that ν( Q) > 0. Hence, we can locate z Q and use our assumption on to conclude that (3.7) ν(3q) ν(b(z, 2 k 0 )) c 2 k 0 = c n diam Q. Therefore, combining (3.6) and (3.7), and writing diam 3Q = 3 diam Q, we obtain (3.8) β2(ν, 2 3Q) diam Q 100 n ( ) 2 k 2 ν(t 3Q). 9c diam Q k=k 1 (k 0 ) T T k (3Q) This estimate is valid for every cube Q 2. quipped with (3.8), we can now bound II. Let l Z be the smallest integer with 2 l r 0. For each k l, write 2 (k) for the family of cubes in 2 with side length 2 k. Then II = β2(ν, 2 3Q) diam Q k 0 =l Q 2 (k 0 ) 100 n 9c k 0 =l Q 2 (k 0 ) k=k 1 (k 0 ) T T k (3Q) ( 2 k n2 k 0 ) 2 ν(t 3Q). Let N 0 = 4 n denote the maximum overlap of cubes 3Q and 3Q where the cubes Q and Q are (closed) dyadic cubes of equal side length. Then II 100N 0 n ( ) 2 k 2 9c n2 k 0 ν(t ) = 100N 0 9c n k 0 =l k=k 1 (k 0 ) T T k k 0 =l k=k 1 (k 0 ) (1/4) k k 0 ν ( Tk ). (Here we used our assumption that the Whitney cubes in T are pairwise disjoint.) Next, set m := k 1 (k 0 ) k 0 = 1 + log 2 3 n. Then, switching the order of summation, II 100N 0 9c n 100N 0 9c n k=l m l+k (l m) k 0 =l k=l m j= m 100N 0 4 m+1 ν(r n \ Γ). 9c n 3 ( ) (1/4) k k 0 ν Tk ( ) (1/4) j ν Tk (Once again we used the disjointness of cubes in T.) Since 4 m log 2 3 n = 16 9n and N 0 = 4 n, it follows that (3.9) II ( n n/c 3 )ν(r n \ Γ). Finally, combining (3.1), (3.2) and (3.9), we conclude that J 2 (ν, r 0, x) dν(x) H 1 (Γ) + ν(r n \ Γ),

12 12 MATTHW BADGR AND RAANAN SCHUL where the implied constant depends only on n and c. More explicitly, (3.10) J 2 (ν, r 0, x) dν(x) ( n)c H 1 (Γ) + ( n n/c 3 ) ν(r n \ Γ), where C is the constant from Corollary 2.3 with a = n and is exponential in the dimension n. We are now ready to prove Theorem A and Corollary B. Proof of Theorem A. Let µ be a locally finite Borel measure on R n and assume that µ is 1-rectifiable. First, because µ is 1-rectifiable, we can find a countable family {Γ i } i=1 of rectifiable curves such that µ gives full mass to i=1 Γ i. Second, because D 1 (µ, ) > 0 µ-a.e. by Lemma 2.7, the measure µ gives full mass to j=1 k=1 j,k, where the set j,k = {x R n : µ(b(x, r)) 2 j r for all r (0, 2 k ]}. Thus, to establish Theorem A, it suffices to prove that J 2 (µ, x) < at µ-a.e. x Γ i j,k for all i, j, k 1. Suppose that Γ = Γ i and = Γ i j,k for some i, j, k 1. Let denote the collection of all dyadic cubes Q in R n such that µ( Q) > 0 and side Q 2 k. Define the measure ν := µ 3Q. First observe that ν has bounded support, since is bounded. Hence ν is finite, because µ is locally finite. Second note that for every x there exists Q x such that x Q x, side Q x = 2 k and B(x, 2 k ) 3Q x. We conclude that ν(b(x, r)) = µ(b(x, r)) 2 j r for all r (0, 2 k ]. Thus J 2 (µ, 2 k, x) dµ(x) = β2(µ, 2 3Q) diam Q χ Q (x) dµ(x) µ(q) Q = β2(ν, 2 3Q) diam Q χ Q (x) dν(x) ν(q) Q = J 2 (ν, 2 k, x) dν(x) H 1 (Γ) + ν(r n \ Γ) <, by Proposition 3.1. In particular, we have J 2 (µ, 2 k, x) < at µ-a.e. x. Therefore, J 2 (µ, x) < at µ-a.e. x, by Lemma 2.9. Proof of the Corollary B. Let µ be a locally finite Borel measure on R n, and assume that µ is 1-rectifiable and µ H 1. On one hand, since µ is 1-rectifiable, we have J 2 (µ, ) < µ-a.e. by Theorem A. On the other hand, since µ H 1, we have D 1 (µ, ) < µ-a.e. by Lemma 2.6. Therefore, J 2 (µ, ) < µ-a.e. by Lemma To wrap up this section, we make two comments about variations of Proposition 3.1, Theorem A and Corollary B and pose an open problem. Remark 3.2. The proof of Proposition 3.1 carries through if, rather than use equation (1.2), we defined the density-normalized Jones function J 2 (µ, r, x) by β2(ν, 2 λq) diam Q ν(q) χ Q(x) side Q r Q

13 MULTISCAL ANALYSIS OF 1-RCTIFIABL MASURS 13 for some λ > 1 arbitrary. Under this scenario, the constant in (3.9) blows up as λ 1. Remark 3.3. In the proof of Proposition 3.1, we assumed that the dyadic cubes used to define the density-normalized Jones function J 2 (µ, r, ) were closed cubes. We could have worked with open or half-open dyadic cubes instead, the only change to the proof being that N 0 = 4 n (closed cubes) improves to N 0 = 3 n (open or half-open cubes). Therefore, Theorem A and Corollary B are true independent of whether the Jones functions are defined using closed, half-open or open dyadic cubes. Problem 3.4. Formulate and prove a version of Theorem A in infinite-dimensional Hilbert space, or show that such a generalization cannot exist. (See Remark 2.4.) 4. Related Work We conclude by discussing some relevant prior work. Recall that the Menger curvature c(x, y, z) of three points x, y, z R n is defined to be the inverse of the radius of the circle that passes through x, y and z. If x, y and z are collinear, then c(x, y, z) = 0. In [Lég99], Léger proved that an integrability condition on Menger curvature is a sufficient test for certain absolutely continuous measures to be 1-rectifiable. Theorem 4.1 ([Lég99]). Suppose R n is Borel, 0 < H 1 () < and µ = H 1. If c 2 (x, y, z) dµ(x)dµ(y)dµ(z) <, then µ is 1-rectifiable. Placed besides one another, Theorem A and Corollary B (necessary conditions) and Theorem 4.1 (a sufficient condition) highlight the importance of curvature to the theory of rectifiability. For an interpretation of beta numbers as a measure of curvature, for the connection between beta numbers and Menger curvature, and for a survey of related results, we refer the reader to [Paj02, Chapter 3] and Schul [Sch07a]. Also see Hahlomaa [Hah08] for a version of Theorem 4.1 that is valid in metric spaces. In [Ler03], Lerman proved that uniform control on an L 2 Jones function ensures that a measure gives positive mass to a rectifiable curve. Moreover, this result is quantitative. To give a precise statement of Lerman s L 2 curve learning theorem, we must introduce a variant of the ordinary Jones function, defined over shifted dyadic grids. Definition 4.2 (Shifted dyadic grids). Redefine the standard dyadic grid (R n ) from above to be the collection of half-open dyadic cubes in R n. For each x R n, let x+ (R n ) denote the shifted dyadic grid that is obtained by translating each cube in (R n ) by x. Define (R n ) to be the union of the 2 n shifted grids x + (R n ), x {0, 1/3} n. Definition 4.3 (L 2 Jones function, Lerman s variant). Let µ be a locally finite Borel measure on R n. An L 2 best fit line for µ in a cube Q is any line l Q which achieves the minimum value of Q dist(x, l Q) 2 dµ(x) among all lines in R n. For every Q (R n ), define ˆβ 2 (µ, Q) by ˆβ 2 (µ, Q) := sup R sup l R Q ( ) 2 dist(x, lr ) dµ(x) diam Q µ(q) where R ranges over all cubes R (R n ) containing Q such that 2 j 0 side R side Q 2j 1, [0, 1]

14 14 MATTHW BADGR AND RAANAN SCHUL and l R ranges over all L 2 best fit lines for µ in R. Here 2 j 0 j 1 are integer parameters. We define the modified L 2 Jones function Ĵ2(µ, r, ) for µ by Ĵ 2 (µ, r, x) = Q ˆβ 2 (µ, Q) 2 χ Q (x) for all x R n, where Q ranges over all cubes in (R n ) of side length at most r > 0. We abbreviate the function Ĵ2(µ, 1, ) starting at scale 1 by Ĵ2(µ, ). We now record Lerman s L 2 curve learning theorem. In the statement of the theorem, spt µ = {x R n : µ(b(x, r)) > 0 for all r > 0} denotes the support of µ. Theorem 4.4 ([Ler03, Theorem 4.8]). Set parameters j 0 = 2 and j 1 = 2 4+log e n (2 18 n < j 1 < 2 19 n). There exist a constant C = C(n) > 1 and an absolute constant λ > 1 with the following property. If µ is a locally finite Borel measure on R n, Q 1 (R n ), and there exists M > 0 such that Ĵ 2 (µ, λ side Q 1, x) M for all x spt µ λq 1, then there exists a rectifiable curve Γ 1 λq 1 such that H 1 (Γ 1 ) Ce CM side Q 1 and µ(γ 1 ) C 1 e CM µ(q 1 ). Remark 4.5. By iterating Theorem 4.4, one can show that uniform control on Ĵ2(µ, ) implies that µ is 1-rectifiable. Let us describe the basic strategy. Suppose that µ is a finite Borel measure supported on Q 0 = (0, 1] n with Ĵ(µ, x) M for all x Q 0 spt µ. Invoking Lerman s theorem once, we find a rectifiable curve Γ 0 that charges a proportion of the µ mass in Q 0. Next divide Q 0 \Γ 0 into Whitney cubes (T i ) i=1, and invoke Lerman s theorem again on each cube T i. This yields a countable family of rectifiable curves (Γ i ) i=1, whose union charges a proportion of the µ mass in Q 0 \Γ 0. To continue, divide Q 0 \ i=0 Γ i into Whitney cubes... and so on. This procedure yields a countable family of rectifiable curves which fully charge the mass of µ. Unfortunately, the proof strategy described in Remark 4.5 cannot be used to show that Ĵ2(µ, x) < at µ-a.e. x R n implies µ is 1-rectifiable. On the other hand, this claim could be proved if one possessed a density version of Lerman s theorem. Conjecture 4.6. A version of Theorem 4.4 holds with the hypothesis Ĵ2(µ, x) M for all x λq 1 spt µ replaced by the set A = {x λq 1 : Ĵ 2 (µ, x) ε} satisfies µ(a) δµ(λq 1 ) for some ε ε 0 (n) and δ δ 0 (n, ε) and the conclusion µ(γ 1 ) C 1 e CM µ(q 1 ) replaced by µ(a Γ 1 ) C 1 e Cε µ(q 1 ) where C = C(n, δ). We believe it should be possible to verify Conjecture 4.6 by rerunning the arguments used by Lerman to prove Theorem 4.4, but we have not checked the details. Finally, we wish to mention a recent paper by Tolsa [Tol12], which introduced the use of tools from the theory of mass transportation to the theory of quantitative rectifiability. In particular, Tolsa established a new characterization of uniformly rectifiable measures, expressed in terms of the L 2 Wasserstein distance W 2 (, ) between probability measures.

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16 16 MATTHW BADGR AND RAANAN SCHUL [Sch07a] Raanan Schul, Ahlfors-regular curves in metric spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 2, MR (2008j:28005) [Sch07b], Subsets of rectifiable curves in Hilbert space the analyst s TSP, J. Anal. Math. 103 (2007), MR (2008m:49205) [Tol12] Xavier Tolsa, Mass transport and uniform rectifiability, Geom. Funct. Anal. 22 (2012), no. 2, MR Department of Mathematics, Stony Brook University, Stony Brook, NY Current address: Department of Mathematics, University of Connecticut, Storrs, CT mail address: matthew.badger@uconn.edu Department of Mathematics, Stony Brook University, Stony Brook, NY mail address: schul@math.sunysb.edu